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Register a new userA note on minimization of rational surfaces obtained from birational dynamical systems
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In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable mappings, including discre
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We prove that the natural $(operatorname{Aff} _2(mathbf{C}),mathbf{C}^2)$-structure on an Inoue surface is the unique $(operatorname{Bir}(mathbb{P}^2),mathbb{P}^2(mathbf{C}))$-structure, generalizing a result of Bruno Klingler which asserts that the natural $(operatorname{Aff}_2(mathbf{C}),mathbf{C}^2)$-structure is the unique $(operatorname{PGL} _3(mathbf{C}),mathbb{P}^2(mathbf{C}))$-structure.
We show how singularities shape the evolution of rational discrete dynamical systems. The stabilisation of the form of the iterates suggests a description providing among other things generalised Hirota form, exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties. We illustrate the phenomenon explicitly with examples covering a wide range of models.
Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded.
Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville--Voisin subring in the Chow groups of Y, and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.
We describe the set of all $(3,1)$-rational functions given on the set of complex $p$-adic field $mathbb C_p$ and having a unique fixed point. We study $p$-adic dynamical systems generated by such $(3,1)$-rational functions and show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We obtain Siegel disks of these dynamical systems. Moreover an upper bound for the set of limit points of each trajectory is given. For each $(3,1)$-rational function on $mathbb C_p$ there is a point $hat x=hat x(f)in mathbb C_p$ which is zero in its denominator. We give explicit formulas of radii of spheres (with the center at the fixed point) containing some points that the trajectories (under actions of $f$) of the points after a finite step come to $hat x$. For a class of $(3,1)$-rational functions defined on the set of $p$-adic numbers $mathbb Q_p$ we study ergodicity properties of the corresponding dynamical systems. We show that if $pgeq 3$ then the $p$-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. For $p=2$, under some conditions we prove non ergodicity and show that there exists a sphere on which the dynamical system is ergodic. Finally, we give a characterization of periodic orbits and some uniformly local properties of the $(3.1)-$rational functions.
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